3.89 \(\int \sec (6 x) \sin (x) \, dx\)

Optimal. Leaf size=85 \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

[Out]

-1/6*arctanh(cos(x)*2^(1/2))*2^(1/2)+1/6*arctanh(2*cos(x)/(1/2*6^(1/2)-1/2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2))
+1/6*arctanh(2*cos(x)/(1/2*6^(1/2)+1/2*2^(1/2)))/(1/2*6^(1/2)+1/2*2^(1/2))

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Rubi [A]  time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4357, 2057, 207, 1166} \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[6*x]*Sin[x],x]

[Out]

-ArcTanh[Sqrt[2]*Cos[x]]/(3*Sqrt[2]) + ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(
2*Cos[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rule 4357

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \sec (6 x) \sin (x) \, dx &=-\operatorname {Subst}\left (\int \frac {1}{-1+18 x^2-48 x^4+32 x^6} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {1}{3 \left (-1+2 x^2\right )}+\frac {4 \left (-1+2 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cos (x)\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\cos (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-8-4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-8+4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\\ \end {align*}

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Mathematica [C]  time = 9.33, size = 627, normalized size = 7.38 \[ \frac {1}{24} \left ((-4-4 i) (-1)^{3/4} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )-1}{\sqrt {2}}\right )-(4-4 i) \sqrt [4]{-1} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+1}{\sqrt {2}}\right )+\frac {2 \left (1+\sqrt {2}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+2}{\sqrt {6}}\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (2 \sin (x)-2 \cos (x)+\sqrt {2}\right )\right )\right )}{2+\sqrt {2}}-\sqrt {2} \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )-2 \sqrt {3} \tanh ^{-1}\left (\frac {\left (\sqrt {2}-1\right ) \tan \left (\frac {x}{2}\right )+\sqrt {2}}{\sqrt {3}}\right )+\log \left (\sec ^2\left (\frac {x}{2}\right ) \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+1\right )\right )\right )+\frac {2 \left (\sqrt {6} \sin (x)+1\right ) \left (\left (2+\sqrt {6}\right ) \sin (x)-\left (\left (2+\sqrt {6}\right ) \cos (x)\right )+\sqrt {6}+3\right ) \left (2 \left (\sqrt {2}+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (2+\sqrt {6}\right ) \tan \left (\frac {x}{2}\right )+2}{\sqrt {2}}\right )+\left (3+\sqrt {6}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (2 \sin (x)-2 \cos (x)+\sqrt {6}\right )\right )\right )\right )}{-2 \left (4 \left (5+2 \sqrt {6}\right ) \sin (x)-6 \sin (2 x)+5 \sqrt {6}+12\right )+\left (12+5 \sqrt {6}\right ) \cos (2 x)+2 \left (5 \sqrt {6} \sin (x)+2 \sqrt {6}+5\right ) \cos (x)}+\frac {\left (\sqrt {2}-2 \sqrt {3} \sin (x)\right ) \left (\left (\sqrt {6}-2\right ) \sin (x)-\left (\left (\sqrt {6}-2\right ) \cos (x)\right )+\sqrt {6}-3\right ) \left (\left (3 \sqrt {2}-2 \sqrt {3}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+\sqrt {3}\right )\right )\right )-2 \left (\sqrt {6}-2\right ) \tanh ^{-1}\left (\left (\sqrt {2}-\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )+\sqrt {2}\right )\right )}{-2 \left (4 \left (2 \sqrt {6}-5\right ) \sin (x)+6 \sin (2 x)+5 \sqrt {6}-12\right )+\left (5 \sqrt {6}-12\right ) \cos (2 x)+2 \left (5 \sqrt {6} \sin (x)+2 \sqrt {6}-5\right ) \cos (x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[6*x]*Sin[x],x]

[Out]

((-4 - 4*I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] - (4 - 4*I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]]
+ (2*(1 + Sqrt[2])*(x + 2*Sqrt[3]*ArcTanh[(2 + (2 + Sqrt[2])*Tan[x/2])/Sqrt[6]] - Log[Sec[x/2]^2] + Log[-(Sec
[x/2]^2*(Sqrt[2] - 2*Cos[x] + 2*Sin[x]))]))/(2 + Sqrt[2]) - Sqrt[2]*(x - 2*Sqrt[3]*ArcTanh[(Sqrt[2] + (-1 + Sq
rt[2])*Tan[x/2])/Sqrt[3]] - Log[Sec[x/2]^2] + Log[Sec[x/2]^2*(1 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x])]) + (2*(2*(
Sqrt[2] + Sqrt[3])*ArcTanh[(2 + (2 + Sqrt[6])*Tan[x/2])/Sqrt[2]] + (3 + Sqrt[6])*(x - Log[Sec[x/2]^2] + Log[-(
Sec[x/2]^2*(Sqrt[6] - 2*Cos[x] + 2*Sin[x]))]))*(1 + Sqrt[6]*Sin[x])*(3 + Sqrt[6] - (2 + Sqrt[6])*Cos[x] + (2 +
Sqrt[6])*Sin[x]))/((12 + 5*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(12 + 5*Sqrt[6
] + 4*(5 + 2*Sqrt[6])*Sin[x] - 6*Sin[2*x])) + ((-2*(-2 + Sqrt[6])*ArcTanh[Sqrt[2] + (Sqrt[2] - Sqrt[3])*Tan[x/
2]] + (3*Sqrt[2] - 2*Sqrt[3])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[3] + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[
x]))]))*(Sqrt[2] - 2*Sqrt[3]*Sin[x])*(-3 + Sqrt[6] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqrt[6])*Sin[x]))/((-12 + 5
*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(-5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(-12 + 5*Sqrt[6] + 4*(-5 + 2*Sqrt[6])*Si
n[x] + 6*Sin[2*x])))/24

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fricas [B]  time = 1.05, size = 153, normalized size = 1.80 \[ -\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + 2 \, \cos \relax (x)\right ) + \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - 2 \, \cos \relax (x)\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + 2 \, \cos \relax (x)\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} - 2 \, \cos \relax (x)\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 1}{2 \, \cos \relax (x)^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(6*x)*sin(x),x, algorithm="fricas")

[Out]

-1/12*sqrt(sqrt(3) + 2)*log(sqrt(sqrt(3) + 2)*(sqrt(3) - 2) + 2*cos(x)) + 1/12*sqrt(sqrt(3) + 2)*log(sqrt(sqrt
(3) + 2)*(sqrt(3) - 2) - 2*cos(x)) + 1/12*sqrt(-sqrt(3) + 2)*log((sqrt(3) + 2)*sqrt(-sqrt(3) + 2) + 2*cos(x))
- 1/12*sqrt(-sqrt(3) + 2)*log((sqrt(3) + 2)*sqrt(-sqrt(3) + 2) - 2*cos(x)) + 1/12*sqrt(2)*log((2*cos(x)^2 - 2*
sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1))

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giac [B]  time = 0.16, size = 182, normalized size = 2.14 \[ -\frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 6 \right |}}\right ) - \frac {2.39014968180000 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 0.0173323801210000\right )}{\frac {268 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 60.0540532247402} + \frac {5.82951931426000 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 0.588790706481000\right )}{\frac {268 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 121.584934401100} + \frac {16.8155413244667 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1.69839637242000\right )}{\frac {268 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 559.622604171000} - \frac {7956.25491093333 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 57.6954805410000\right )}{\frac {268 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 168981.261592000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(6*x)*sin(x),x, algorithm="giac")

[Out]

-1/12*sqrt(2)*log(abs(-4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)/abs(4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1
) - 6)) - 2.39014968180000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.0173323801210000)/(268*(cos(x) - 1)/(cos(x) + 1)
+ 60.0540532247402) + 5.82951931426000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.588790706481000)/(268*(cos(x) - 1)/
(cos(x) + 1) + 121.584934401100) + 16.8155413244667*log(-(cos(x) - 1)/(cos(x) + 1) - 1.69839637242000)/(268*(c
os(x) - 1)/(cos(x) + 1) + 559.622604171000) - 7956.25491093333*log(-(cos(x) - 1)/(cos(x) + 1) - 57.69548054100
00)/(268*(cos(x) - 1)/(cos(x) + 1) - 168981.261592000)

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maple [A]  time = 0.24, size = 80, normalized size = 0.94 \[ \frac {2 \arctanh \left (\frac {8 \cos \relax (x )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}-2 \sqrt {2}\right )}+\frac {2 \arctanh \left (\frac {8 \cos \relax (x )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}+2 \sqrt {2}\right )}-\frac {\arctanh \left (\cos \relax (x ) \sqrt {2}\right ) \sqrt {2}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(6*x)*sin(x),x)

[Out]

2/3/(2*6^(1/2)-2*2^(1/2))*arctanh(8*cos(x)/(2*6^(1/2)-2*2^(1/2)))+2/3/(2*6^(1/2)+2*2^(1/2))*arctanh(8*cos(x)/(
2*6^(1/2)+2*2^(1/2)))-1/6*arctanh(cos(x)*2^(1/2))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{24} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) + 2 \, {\left (\sqrt {2} \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 1\right ) + \frac {1}{24} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) - 2 \, {\left (\sqrt {2} \cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 1\right ) - \int \frac {{\left (\sin \left (7 \, x\right ) - \sin \left (5 \, x\right ) + \sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (8 \, x\right ) - {\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (7 \, x\right ) - \cos \left (5 \, x\right ) + \cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (8 \, x\right ) - {\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (7 \, x\right ) + {\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) + {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) + \cos \left (7 \, x\right ) \sin \left (4 \, x\right ) - \cos \left (5 \, x\right ) \sin \left (4 \, x\right ) + \sin \left (3 \, x\right ) - \sin \relax (x)}{3 \, {\left (2 \, {\left (\cos \left (4 \, x\right ) - 1\right )} \cos \left (8 \, x\right ) - \cos \left (8 \, x\right )^{2} - \cos \left (4 \, x\right )^{2} - \sin \left (8 \, x\right )^{2} + 2 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) - \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(6*x)*sin(x),x, algorithm="maxima")

[Out]

-1/24*sqrt(2)*log(2*sqrt(2)*sin(2*x)*sin(x) + 2*(sqrt(2)*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(
2*x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 1) + 1/24*sqrt(2)*log(-2*sqrt(2)*sin(2*x)*sin(x) - 2*(sqrt(2)*cos(x)
- 1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(2*x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) + 1) - integrate(1/3*((si
n(7*x) - sin(5*x) + sin(3*x) - sin(x))*cos(8*x) - (sin(3*x) - sin(x))*cos(4*x) - (cos(7*x) - cos(5*x) + cos(3*
x) - cos(x))*sin(8*x) - (cos(4*x) - 1)*sin(7*x) + (cos(4*x) - 1)*sin(5*x) + (cos(3*x) - cos(x))*sin(4*x) + cos
(7*x)*sin(4*x) - cos(5*x)*sin(4*x) + sin(3*x) - sin(x))/(2*(cos(4*x) - 1)*cos(8*x) - cos(8*x)^2 - cos(4*x)^2 -
sin(8*x)^2 + 2*sin(8*x)*sin(4*x) - sin(4*x)^2 + 2*cos(4*x) - 1), x)

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mupad [B]  time = 2.28, size = 118, normalized size = 1.39 \[ \mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\cos \relax (x)}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}+\frac {3\,\sqrt {6}\,\cos \relax (x)}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\cos \relax (x)}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}-\frac {3\,\sqrt {6}\,\cos \relax (x)}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}-\frac {\sqrt {6}}{12}\right )-\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\cos \relax (x)\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/cos(6*x),x)

[Out]

atanh((5*2^(1/2)*cos(x))/(2097152*((2^(1/2)*6^(1/2))/4194304 + 1/1048576)) + (3*6^(1/2)*cos(x))/(2097152*((2^(
1/2)*6^(1/2))/4194304 + 1/1048576)))*(2^(1/2)/12 + 6^(1/2)/12) - atanh((5*2^(1/2)*cos(x))/(2097152*((2^(1/2)*6
^(1/2))/4194304 - 1/1048576)) - (3*6^(1/2)*cos(x))/(2097152*((2^(1/2)*6^(1/2))/4194304 - 1/1048576)))*(2^(1/2)
/12 - 6^(1/2)/12) - (2^(1/2)*atanh(2^(1/2)*cos(x)))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\relax (x )} \sec {\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(6*x)*sin(x),x)

[Out]

Integral(sin(x)*sec(6*x), x)

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